Horizon (1964) Episode Scripts

N/A - Fermat's Last Theorem

1
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METRONOME TICKS
Perhaps I could best describe
my experience of doing mathematics
in terms of entering a dark mansion.
One goes into the first room
and it's dark - completely dark.
One stumbles around
bumping into the furniture.
And gradually you learn
where each piece of furniture is
and finally, after six months or so,
you find the light switch,
you turn it on
and suddenly it's all illuminated.
You can see exactly where you were.
At the beginning of September
I was sitting here at this desk
when suddenly, totally unexpectedly,
I had this incredible revelation.
It was the most
The most important moment
of my working life.
Nothing I ever do again will
I'm sorry.
This is the story
of one man's obsession
with the world's greatest
mathematical problem.
For seven years Professor Andrew
Wiles worked in complete secrecy,
creating the calculation
of the century.
It was a calculation which
brought him fameand regret.
So I came to this.
I was a ten-year-old.
And one day I happened to be
looking in my local public library
and I found a book on math.
And it told a bit about
the history of this problem.
That someone had resolved
this problem 300 years ago,
but no-one had ever seen the proof,
no-one knew if there WAS a proof.
People ever since
had looked for the proof.
Here was a problem that I,
a ten-year-old, could understand,
but none of the great mathematicians
in the past had been able to resolve.
And, from that moment, of course,
I just tried to solve it myself.
It was such a challenge -
such a beautiful problem.
This problem was
Fermat's last theorem.
'Pierre de Fermat was
a 17th-century French mathematician
'who made some of
the greatest breakthroughs
'in the history of numbers.'
'His inspiration came
from studying the Arithmetica,
'an ancient Greek text.'
Fermat owned a copy of this book,
which is a book about numbers
with lots of problems, which
presumably Fermat tried to solve.
He studied it,
he wrote notes in the margins.
'Fermat's original notes were lost,
'but they can still be read
in a book published by his son.
'It was one of these notes
that was Fermat's greatest legacy.'
And his is the fantastic observation
of Master Pierre de Fermat
which caused all the trouble.
"Cubum autem in duos cubos"
'This tiny note is the world's
hardest mathematical problem.
'It's been unsolved for centuries,
'yet it begins with
an equation so simple
'that children know it off by heart.'
ALL: The square of the hypotenuse
is equal to the sum of
the squares of the other two sides.
Yeah, well, that's
Pythagoras' theorem, isn't it?
That's what we all did at school.
So, Pythagoras' theorem
The clever thing about it
is that it tells us
when three numbers are the sides of
a right-angle triangle.
That happens just when x squared
plus y squared equals z squared.
WILES: x squared plus y squared
equals z squared.
And you can ask, well,
what are the whole number solutions
of this equation?
And you quickly find
there's a solution:
3 squared plus 4 squared
equals 5 squared.
Another one is 5 squared
plus 12 squared is 13 squared.
And you go on looking
and you find more and more.
So then a natural question is -
the question Fermat raised -
supposing you change from squares?
Supposing you replace
the two by three,
by four, by five, by six -
by any whole number "n"?
And Fermat said simply
that you'll never find
any solutions -
however far you look,
you'll never find a solution.
'You will never find numbers
that fit this equation
'if n is greater than 2.
'That's what Fermat said.
And, what's more,
'he said he could prove it.
'In a moment of brilliance,
'he scribbled the following
mysterious note.'
Written in Latin, he says
he has a truly wonderful proof -
"demonstrationem mirabilem" -
of this fact.
And then last words are
"hanc marginis exiguitas
non caperet" -
"this margin is too small
to contain it."
'So Fermat said he had a proof,
'but he never said what it was.'
Fermat made LOTS of marginal notes.
People took them as challenges.
And over the centuries every single
one of them has been disposed of
and the last one
to be disposed of is this one -
that's why it's called
"the last theorem".
Rediscovering Fermat's proof
became the ultimate challenge -
a challenge which
would baffle mathematicians
for the next 300 years.
Gauss, the greatest mathematician
in the world.
Oh, yes, Galois.
Kummer, of course.
Well, in the 18th century,
Euler didn't prove it.
Well, you know, there's only been
the one woman, really.
Sophie Germain.
There are millions of
Lots of people.
But nobody had any idea
where to start.
Well, mathematicians
just love a challenge,
and this problem, this particular
problem just looked so simple,
it just looked as if
it had to have a solution.
And, of course, it's very special
because Fermat said
he HAD a solution.
'Mathematicians had to prove
that no numbers fitted this equation.
'But with the advent of computers,
'couldn't they check
each number one by one
'and show that none of them fitted?'
Well, how many numbers are there
to be dealt with?
You've got to do it
for infinitely many numbers.
So after you've done it for one,
how much closer have you got?
Well, there's still
infinitely many left.
After you've done it
for 1,000 numbers -
how much closer have you got?
Well, there's still
infinitely many left.
After you've done it for a million,
there's still infinitely many left -
in fact, you haven't done very many,
have you?
'A computer can never
check every number.
'Instead, what's needed
is a mathematical proof.'
A mathematician is not happy
until the proof is complete
and considered complete by
the standards of mathematics.
In mathematics there's
the concept of proving something,
of knowing it with
absolute certainty.
Which Well, it's called
"rigorous proof".
Well, a rigorous proof
is a series of arguments
..based on logical deductions
..which just build
one upon another
..step by step
..until you get to
..a complete proof.
That's what mathematics is about.
'A proof is a sort of reason.
'It explains why
no numbers fit the equation
'without having to check
every number.
'After centuries of
failing to find a proof,
'mathematicians began
to abandon Fermat
'in favour of more serious maths.'
MUSIC: "Metal Guru"
by T Rex
'In the '70s
Fermat was no longer in fashion.
'At the same time
'Andrew Wiles was just beginning
his career as a mathematician.'
'He went to Cambridge
as a research student
'under the supervision
of Professor John Coates.'
I've been very fortunate
to have Andrew as a student,
and even as a research student,
he, erm He was
a wonderful person to work with.
He had very deep ideas then
and it was always clear
he was a mathematician
who would do great things.
'But not with Fermat.
'Everyone thought Fermat's
last theorem was impossible,
'so Professor Coates encouraged
Andrew to forget his childhood dream
'and work on more mainstream maths.'
WILES: The problem with
working on Fermat
is that you could
spend years getting nothing.
So when I went to Cambridge
my advisor, John Coates,
was working on Iwasawa theory
and elliptic curves
and I started working with him.
'Elliptic curves were
the "in" thing to study,
'but, perversely, elliptic curves
'are neither ellipses nor curves.'
You may not have heard
of elliptic curves,
but they're extremely important.
OK, so what's an elliptic curve?
Elliptic curves -
they're not ellipses.
They're cubic curves
whose solution have a shape
that looks like a doughnut.
It looks so simple
yet the complexity,
especially arithmetic complexity,
is immense.
'Every point on the doughnut
is the solution to an equation.
'Andrew Wiles now studied
these elliptic equations
'and set aside his dream.'
'What he didn't realise was that
on the other side of the world.
'elliptic curves
and Fermat's last theorem
'were becoming inextricably linked.'
BELL TOLLS
'I entered the University of Tokyo'
in 1949.
That was four years after the war,
but almost all the professors
weretired
and the lectures were not inspiring.
'Goro Shimura and his fellow students
'had to rely on each other
for inspiration.
'In particular, he formed
a remarkable partnership
'with a young man
by the name of Yutaka Taniyama.'
That was when I became
very close to Taniyama.
Taniyama was not
a very careful person
as a mathematician.
He made a lot of mistakes.
But he
He made mistakes in a good direction
and so, eventually,
he got right answers.
And I tried to imitate him,
but I found out that
it is very difficult
to make good mistakes.
'Together, Taniyama and Shimura
worked on the complex mathematics
'of modular functions.'
I really can't explain what a modular
function is in one sentence.
I can try and give you
a few sentences to explain.
HE LAUGHS
I really can't do it in one sentence.
Oh, it's impossible!
There's a saying
attributed to Eichler
..that there are five fundamental
operations of arithmetic -
addition, subtraction,
multiplication,
division, and modular forms.
Modular forms are
functions on the complex plane
that are inordinately symmetric.
They satisfy so many
internal symmetries
that their mere existence
seem like accidents,
but they DO exist.
'This image is merely
a shadow of a modular form.
'To see one properly,
'your TV screen
would have to be stretched
'into something called
hyperbolic space.'
'Bizarre modular forms
seem to have nothing whatsoever
'to do with the humdrum world
of elliptic curves.'
'But what Taniyama and Shimura
suggested shocked everyone.'
In 1955 there was
an international symposium
and Taniyama posed
two or three problems.
'The problems posed by Taniyama
'led to the extraordinary claim
'that every elliptic curve
'was really a modular form
in disguise.'
'It became known as
the Taniyama-Shimura conjecture.'
The Taniyama-Shimura conjecture
says that every rational
elliptic curve is modular,
and that's SO hard to explain.
BARRY MAZUR: So let me explain.
Over here you have
the elliptic world -
the elliptic curves, these doughnuts,
and over here
you have the modular world -
modular forms
with their many, many symmetries.
The Shimura-Taniyama conjecture
makes a bridge
between these two worlds -
these worlds live on
different planets.
It's a bridge. It's more than
a bridge - it's really a dictionary.
A dictionary where questions,
intuitions, insights, theorems
in the one world
get translated to
questions, intuitions
in the other world.
I think that when
Shimura and Taniyama
first started talking about
the relationship
between elliptic curves
and modular forms,
people were very incredulous.
I wasn't studying mathematics yet.
By the time I was a graduate student,
in 1969 or 1970,
people were coming
to believe the conjecture.
'In fact, Taniyama-Shimura became
a foundation for other theories,
'which all came to depend on it.
'But Taniyama-Shimura was only
a conjecture, an unproven idea,
'and until it could be proved,
'all the maths which relied on it
was under threat.'
WILES: We built
more and more conjectures,
stretched further and further
into the future
but they would all be
completely ridiculous
if Taniyama-Shimura was not true.
'Proving the conjecture
became crucial,
'but, tragically,
the man whose idea inspired it
'didn't live to see
the enormous impact of his work.
'In 1958,
Taniyama committed suicide.'
I was very much puzzled.
Puzzlement may be the best word.
Of course I was sad, but
See, it was so sudden,
andI was unable
to make sense out of this.
'Taniyama-Shimura went on to become
'one of the great
unproven conjectures.
'But what did it have to do
with Fermat's last theorem?'
At that time,
no-one had any idea
that Taniyama-Shimura
could have anything to do
with Fermat.
Of course, in the '80s,
that all changed completely.
MUSIC: "One Way Or Another"
by Blondie
'Taniyama-Shimura says
"every elliptic curve is modular",
'and Fermat says
"no numbers fit this equation".
'What was the connection?'
# One way or another
I'm gonna find ya
# I'm gonna get ya
Get ya, get ya, get ya
# One way or another
I'm gonna win ya
# I'm gonna get ya
Get ya, get ya, get ya
# One way or another
# I'm gonna see ya
# I'm gonna meet ya
Meet ya, meet ya, meet ya
One day
On the face of it,
the Shimura-Taniyama conjecture,
which is about elliptic curves,
and Fermat's last theorem
have nothing to do with each other -
because there's no connection
between Fermat and elliptic curves.
But in 1985, Gerhard Frey
had this amazing idea.
'Frey, a German mathematician,
'considered the unthinkable -
'what would happen if Fermat
was wrong,
'and there WAS a solution
to this equation after all?'
Frey showed how,
starting with a fictitious solution
to Fermat's last equation,
if indeed
such a horrible beast existed,
he could make an elliptic curve
with some very weird properties.
That elliptic curve
seems to be not modular,
but Shimura-Taniyama says that
every elliptic curve IS modular.
'So if there IS a solution
to this equation,
'it creates
such a weird elliptic curve
'it defies Taniyama-Shimura.'
So, in other words, if Fermat
is false, so is Shimura-Taniyama.
Or, said differently,
if Shimura-Taniyama is correct,
so is Fermat's last theorem.
'Fermat and Taniyama-Shimura
were now linked -
'apart from just one thing.'
The problem is that
Frey didn't really PROVE
that his elliptic curve
was not modular.
He gave a plausibility argument
which he hoped
could be filled in by experts,
and then the experts
started working on it.
'In theory, you could prove Fermat
by proving Taniyama -
'but only if Frey was right.
'Frey's idea became known
as the epsilon conjecture,
'and everyone tried to check it.'
'One year later, in San Francisco,
'there was a breakthrough.'
KEN RIBET: I saw Barry Mazur
on the campus
and I said,
"Let's go for a cup of coffee,"
and we sat down
for cappuccinos at this cafe.
And I looked at Barry and I said,
"You know, I'm trying
to generalise what I've done
"so that we can prove
the full strength
"of Serre's epsilon conjecture."
And Barry looked at me and said,
"But you've done it already!
"All you have to do is add on some
extra gamma-zero of (M) structure,
"and run through your argument
and it still works,
"and that gives
everything you need."
And this had never occurred to me,
as simple as it sounds.
I looked at Barry, I looked at
my cappuccino, I looked back at Barry
and I said,
"My God, you're absolutely right!"
Ken's idea was brilliant!
I was at a friend's house sipping
iced tea early in the evening
and he just mentioned casually
in the middle of a conversation,
"By the way, did you hear that Ken
has proved the epsilon conjecture?"
And I was just electrified.
I knew that moment
..the course of my life was changing
because this meant that
..to prove Fermat's last theorem,
I just had to prove
the Taniyama-Shimura conjecture.
From that moment,
that was what I was working on.
I just knew I would go home
and work on
the Taniyama-Shimura conjecture.
'Andrew abandoned
all his other research.
'He cut himself off
from the rest of the world
'and for the next seven years
'he concentrated solely
on his childhood passion.'
WILES: I never use a computer.
I sometimes
I scribble, I do doodles.
I start trying to
To find patterns, really,
so I'm doing calculations
which try to explain
some little piece of mathematics
and I'm trying to fit it in with
some previous, broad
conceptual understanding of
some branch of mathematics.
Sometimes that'll involve going
and looking up in a book
to see how it's done there.
Sometimes it's a question
of modifying things a bit,
sometimes doing
a little extra calculation.
And sometimes you realise
that nothing that's ever
been done before is any use at all.
And you just have to find
something completely new.
Andit's a mystery
where it comes from.
'I must confess I did not think that
the Shimura-Taniyama conjecture'
was accessible to proof at present,
I thought I probably wouldn't
see a proof in my lifetime.
I was one of the
vast majority of people
who believed that
the Shimura-Taniyama conjecture
was just completely inaccessible,
and I didn't bother to prove it,
even think about trying to prove it.
Andrew Wiles is probably
one of the few people on earth
who had the audacity to dream
that you can actually
go and prove this conjecture.
In this case,
certainly the first several years,
I had no fear of competition.
I simply didn't think I,
or anyone else,
hadany real idea how to do it.
But I realised after a while
that talking to people casually
about Fermat was impossible.
Because it just generates
too much interest
and you can't really
focus yourself for years
unless you have this kind of
undivided concentration,
which too many spectators
would have destroyed.
'Andrew decided that he would work
in secrecy and isolation.'
I often wondered to myself
what he was working on!
Didn't have an inkling.
No, I suspected nothing.
This is probably the only case I know
where someone worked
for such a long time
without divulging what he was doing,
without talking about
the progress he had made.
It's just unprecedented.
METRONOME TICKS
'Andrew was embarking on one of the
most complex calculations in history.
'For the first two years,
'he did nothing but
immerse himself in the problem,
'trying to find a strategy
which might work.'
So it was now known
that Taniyama-Shimura
implied Fermat's last theorem.
What does Taniyama-Shimura say?
It says that all elliptic curves
should be modular.
Well, this was an old problem
that had been around for 20 years
and lots of people
would have tried to solve it.
One way of looking at it is
that you have ALL elliptic curves,
and then you have
the MODULAR elliptic curves.
You want to prove that
there are the same number of each.
Now, of course,
you're talking about infinite sets
so you can't just count them per se,
but you can divide them into packets
and you could try to count
each packet and see how things go.
And this proves to be a very
attractive idea for about 30 seconds
but you can't really get
much further than that.
And the big question on the subject
was how you could possibly count.
And, in effect, Wiles introduced
the correct technique.
'Andrew's trick was
to transform the elliptic curves
'into something called
Galois representations
'which would make counting easier.
'Now it was a question
of comparing modular forms
'with Galois representations,
'not elliptic curves.'
Now, you might ask -
and it's an obvious question -
why can't you do this with
elliptic curves and modular forms?
Why couldn't you count
elliptic curves, count modular forms,
show they're the same number?
Well, the answer is, people tried,
and they never found
a way of counting them.
And this was why
this is the key breakthrough -
that I had found a way to count,
not the original problem,
but the modified problem.
I'd found a way to count modular
forms and Galois representations.
'This was only the first step,
'and already it had taken
three years of Andrew's life.'
My wife's only known me
while I've been working on Fermat.
I told her a few days
after we got married.
I decided that I really only had time
for my problem and my family.
And when I was
concentrating very hard,
then I found that,
with young children,
that's the best possible way
to relax.
When you're talking to young children
they simply aren't interested
in Fermat -
at least at this age.
They want to hear a children's story
and they're not going to let you
do anything else.
So I'd found this wonderful
counting mechanism
and I started thinking
about this concrete problem
in terms of Iwasawa theory.
Iwasawa theory was the subject
I'd studied as a graduate student
and, in fact, with my advisor,
John Coates,
I'd used it
to analyse elliptic curves.
'Andrew hoped that Iwasawa theory
'would complete
his counting strategy.'
Now, I tried to use
Iwasawa theory in this context,
but I ran into trouble.
I seemed to be up against a wall.
I just didn't seem to
be able to get past it.
Well, sometimes when
I can't see what to do next, I
often come here by the lake.
Walking has a very good effect
in that you're in this state of
concentration,
but at the same time you're relaxing,
you're allowing the subconscious
to work on you.
'Iwasawa theory
was supposed to help create
'something called
a class number formula.
'But several months passed
'and the class number formula
remained out of reach.'
So at the end of the summer of '91
I was at a conference.
John Coates told me
about a wonderful new paper
of Matthias Flach,
a student of his,
in which he had tackled
the class number formula,
in fact, exactly
the class number formula I needed.
So Flach,
using ideas of Kolyvagin
..had made
a very significant first step
in actually producing
the class number formula.
So at that point I thought,
"This is just what I need,
this is tailor-made for the problem."
I put aside completely
the old approach I'd been trying
and I devoted myself day and night
to extending his result.
'Andrew was almost there,
'but this breakthrough
was risky and complicated.
'After six years of secrecy,
he needed to confide in someone.'
January of 1993,
Andrew came up to me one day at tea,
asked me if I could
come up to his office,
there was something
he wanted to talk to me about.
I had no idea what this could be.
I went up to his office,
he closed the door,
he said he thought he would be able
to prove Taniyama-Shimura.
I was just amazed.
This was fantastic.
It involved a kind of mathematics
that Nick Katz is an expert in.
I think another reason
he asked me was
..that he was sure
I would not tell other people.
I would keep my mouth shut -
which I did.
JOHN CONWAY: 'Andrew Wiles
and Nick Katz
'had been spending
rather a lot of time'
huddled over a coffee table
at a far end of the common room
working on some problem or other.
We never knew what it was.
'In order not to arouse
any more suspicion,
'Andrew decided to check his proof
'by disguising it
in a course of lectures
'which Nick Katz could then attend.'
Well, I explained
at the beginning of the course
that Flach had
written this beautiful paper
and I wanted to try to extend it
to prove the full
class number formula.
The only thing I didn't explain
was that proving
the class number formula
was most of the way
to Fermat's last theorem.
So this course was announced
that said "Calculations On Elliptic
Curves", which could mean anything.
Didn't mention Fermat,
didn't mention Taniyama-Shimura -
there was no way in the world
anyone could have guessed
that it was about that
if he didn't already know.
None of the graduate students knew,
and in a few weeks they just drifted
off because it's impossible
to follow stuff if you don't know
what it's for, pretty much.
It's pretty hard even if
you DO know what it's for.
But after a few weeks,
I was the only guy in the audience.
'The lectures revealed no errors
'and still none of
his colleagues suspected
'why Andrew was being so secretive.'
Maybe he's run out of ideas!
That's why he's quiet -
you never know why they're quiet.
'The proof was still missing
a vital ingredient,
'but Andrew now felt confident.
'It was time to tell
one more person.'
I called up Peter
and asked him
if I could come round and
talk to him about something.
I got a phone call from Andrew
saying that he had
something very important
he wanted to chat to me about.
And, sure enough,
he had some very exciting news.
I said, "I think you'd better
sit down for this." And he sat down.
I said, "I think I'm about to prove
Fermat's last theorem."
I was flabbergasted.
Excited. Disturbed.
I mean, I remember that night
finding it quite difficult to sleep.
METRONOME TICKS
But there was still a problem.
Late in the spring of '93,
I was in this very awkward position.
I thought I'd got most of
the curves being modular -
so that was NEARLY enough
to be content
to have Fermat's last theorem -
but there was this
These few families of elliptic curves
that had escaped the net.
I was sitting here at my desk
in May of '93,
still wondering about this problem.
And I was casually glancing
at a paper of Barry Mazur's
and there was just one sentence
which made a reference to actually
what is a 19th-century construction.
And I just instantly realised that
there was a trick that I could use,
that I could switch from the families
of elliptic curves I'd been using.
I'd been studying them
using the prime three -
I could switch and study them
using the prime five.
It looked more complicated,
but I could switch from
these awkward curves
that I couldn't prove were modular
to a different set of curves
which I'd already proved
were modular,
and use that information
to just go that one last step.
And
I just kept working out the details
and time went by
and I forgot to go down to lunch
and it got to about teatime.
And I went down and Nada was very
surprised that I'd arrived so late.
Then she I told her that I
Ibelieved I'd solved
Fermat's last theorem.
I was convinced that
I had Fermat in my hands,
and there was a conference
in Cambridge
organised by my advisor, John Coates.
I thought that would be
a wonderful place.
It's my old home town.
I'd been a graduate student there.
It would be a wonderful place
to talk about it
if I could get it in good shape.
The name of the lectures
that he announced was simply
Elliptic Curves And Modular Forms.
There was no mention
of Fermat's last theorem.
Well, I was at this conference
on L functions and elliptic curves,
it was kind of a standard conference
and all of the people were there,
didn't seem to be
anything out of the ordinary.
And so people started telling me
that they'd been hearing
weird rumours
about Andrew Wiles's
proposed series of lectures.
I started talking to people
and I got more and more
precise information.
I've no idea how it was spread.
Not from me. Not from me.
'Whenever any piece of mathematical
news had been in the air'
Peter would say, "Oh, that's nothing.
"Wait until you hear the BIG news.
"There's something big
going to break."
Maybe some hints, yeah.
People would ask me,
leading up to my lectures,
what exactly I was going to say.
And I said,
"Well, come to my lecture and see."
A sort of very charged atmosphere.
A lot of the major figures
in arithmetical, algebraic geometry
were there -
Richard Taylor,
John Coates, Barry Mazur.
Well, I'd never seen a lecture series
in mathematics like that before.
What was unique about those lectures
were the glorious ideas,
how many new ideas were presented,
and the constancy
of his dramatic build-up -
it was suspenseful until the end.
'There was this marvellous moment'
when we were coming close to
a proof of Fermat's last theorem -
the tension had built up
and there was only
one possible punchline.
So after I'd explained the
3 - 5 switch on the blackboard,
I then just wrote up
a statement of Fermat's last theorem,
said I'd proved it,
said, "I think I'll stop there."
TUMULTUOUS APPLAUSE
'The next day,
what was totally unexpected'
was that we were deluged
by enquiries from
newspapers, journalists
from all around the world.
It was a wonderful feeling
after seven years
to have really
solved my problem.
I had finally done it.
Only later did itcome out
that there was a problem at the end.
Now it was time
for it to be refereed -
which is to say, for people
appointed by the journal
to go through and make sure
that the thing was really correct.
So for two months, July and August,
I literally did nothing
but go through this manuscript
line by line.
What this meant concretely was that
essentially every day,
sometimes twice a day,
I would e-mail Andrew
with a question -
"I don't understand what you say
on this page, on this line,
"it seems to be wrong,"
or "I just don't understand."
So Nick was sending me e-mails
and at the end of the summer,
he sent one that
seemed innocent at first
and I tried to resolve it.
It's a little bit complicated,
so he sends me a fax.
But the fax doesn't seem to answer
the question, so I e-mail him back
and I get another fax
which I'm still not satisfied with.
And this, in fact,
turned into the error
that turned out to be
a fundamental error
and that we had completely missed
when he was lecturing in the spring.
That's where the problem was -
in the method of Flach and Kolyvagin
that I'd extended.
So once I realised that,
at the end of September, that
..there was really a problem
with the way
I'd made the construction,
I spent the fall trying to
think what kind of modifications
could be made to the construction.
There are lots of simple
and rather natural modifications,
any one of which might have worked.
And every time he would
try and fix it in one corner,
some other difficulty
would add up in another corner.
It was like he was
trying to put a carpet
in a room where the carpet
had more size than the room.
He could put it in in any corner,
and then when
he ran to the other corner
it would pop up in this corner.
Whether you could not
put the carpet in the room
was not something
that he was able to decide.
NICK KATZ: I think
he externally appeared normal
but at this point he was
..keeping a secret from the world.
AndI think he must have been,
in fact,
pretty uncomfortable about it.
Well, you know, we were behaving
a little bit like criminologists.
Nobody actually liked to come out
and ask him
how he's getting on with the proof,
so somebody would say,
"I saw Andrew this morning."
"Did he smile?"
"Well, yes, but he didn't
look TOO happy."
The first seven years
I'd worked on this problem,
I loved every minute of it.
However hard it had been -
there'd been setbacks often,
there'd been things that
had seemed insurmountable,
but it was a kind of private
umand very personal battle
I was engaged in.
And then, after there was
a problem with it
Doing mathematics in that kind of
..rather over-exposed way
is certainly not my style
and I have no wish to repeat it.
METRONOME TICKS
'In desperation,
Andrew called for help.
'He invited his former student,
Richard Taylor,
'to work with him in Princeton.
'But no solution came.
'After a year of failure, he was
ready to abandon his flawed proof.'
In September,
I
decided to go back
and look one more time
at theoriginal
structure of Flach and Kolyvagin
to try and pinpoint exactly
why it wasn't working,
try and formulate it precisely.
One can never really do that
in mathematics,
but I just wanted to
set my mind at rest
that it really couldn't
be made to work.
And I was sitting here, at this desk.
It was a Monday morning,
19th September.
And I was trying
convincing myself
that it didn't work,
just seeing exactly
what the problem was,
when suddenly,
totally unexpectedly,
I had this incredible revelation.
I
I realised what washolding me up
was exactly what would
resolve the problem I'd had
in my Iwasawa theory attempt
three years earlier.
It was
It was the most, er
The most important moment
of my working life.
It was so indescribably beautiful.
It was so simple and so elegant.
AndI just
stared in disbelief for20 minutes.
And then, during the day,
I walked round the department,
I'd keep coming back to my desk and
looking to see it was still there,
and it was still there.
Almost what seemed to be stopping
the method of Flach and Kolyvagin
was exactly what would make
horizontal Iwasawa theory.
My original approach to the problem
from three years before
would make exactly THAT work.
So out of the ashes seemed to rise
the true answer to the problem.
Sothe first night
I went back and slept on it.
I checked through it again
the next morning
and by 11 o'clock I was satisfied
and I went down and told my wife,
"I've got it.
I think I've got it - I've found it."
And it was so unexpected,
I think she thought I was talking
about a children's toy or something.
She said, "Got what?"
And I said,
"I've fixed my proof. I've got it."
I think it will always stand
as one of the high achievements
of number theory.
It was magnificent.
It's not every day that you hear
the proof of the century.
Well, my first reaction was
"I told you so!"
The Taniyama-Shimura conjecture
is no longer a conjecture,
and, as a result,
Fermat's last theorem
has been proved.
But is Andrew's proof
the same as Fermat's?
Fermat couldn't possibly
have had this proof.
It's a 20th-century proof,
there's no way this could have
been done before the 20th century.
I'm relieved that this result
is now settled.
But I'm sad in some ways
because Fermat's last theorem
has been responsible for so much.
What will we find to take its place?
There's no other problem
that will mean the same to me.
I had this very rare privilege
of being able to pursue
..in my adult life,
what had been my childhood dream.
I I know it's a rare privilege,
but if one can do this
it's more rewarding
than anything I could imagine.
One of the great things
about this work
is that it embraces the ideas
of so many mathematicians.
I've made a partial list -
Klein, Fricke,
Hurwitz, Hecke,
Dirichlet, Dedekind
The proof by Langlands and Tunnell
Deligne, Rapoport, Katz
Mazur's idea of using the deformation
theory of Galois representations
..Igusa, Eichler,
Shimura, Taniyama
Frey's reduction
The list goes on and on
..Bloch, Kato,
Selmer, Frey,
Fermat.